4.2 Extension of the atom chip setup
4.2.1 Michelson interferometer for membrane readout
The amplitude of the membrane oscillator is read out with a Michelson interferom- eter as shown in Fig. 4.3. The interferometer consists of a beam splitter (BS) which splits a light beam from a coherent light source (LS) into two beams which are each reflected at a mirror (M1, M2). The reflected beams are overlapped at the beam splitter, and the power is measured with a photodetector (PD).
The power incident on the photodiode depends on the difference of the optical path lengths of the two interferometer arms with optical path lengths L1 and L2. The phase difference is given [122] by
φ= 2π(2(L1−L2)/λ) (4.51)
where λ is the wavelength of the light. The power incident on the photodetector is given by
P = 1
2αP0(1 +Csinφ), (4.52)
with the optical powerP0 of the light source. C models the contrast due to unequal
powers in L1 and L2. α is an attentuation factor which accounts for optical losses in the interferometer which occur e.g. due to reflections at air-glass interfaces. The power shows a sine dependence as indicated in Fig. 4.3 (right), if e.g. L1 is fixed
4.2 Extension of the atom chip setup 61
Figure 4.3: (left) Michelson interferometer. (right) Power modulation at the output port for variation of the length of one arm.
an adjustment on the steepest slope. The interferometer measures the amplitude of the membrane. The oscillation amplitude is linearly translated into an oscillating voltage by the photodiode, and can be analyzed with e.g. a Lock-In amplifier. One fundamental limit to the sensitivity is shotnoise in the photodetector (PD). The poisson distributed shotnoise of a currentI has a standard deviation of
σI =
p
hi2i=p2eI∆f , (4.53)
with the electron charge e and the measurement bandwidth ∆f. This has to be compared to the change of the PD current I +δI due to a length change of one interferometer arm.
Setup of a Michelson interferometer
The experimental setup of the Michelson interferometer is shown in Fig. 4.4. A home built, free running diode laser atλ= 830nm is used as a coherent light source. A grating provides feedback into the laserdiode for narrowing the linewidth [123], and is adjusted for single mode operation. The beam is shaped with an anamorphic prism pair (AP) and a telescope, and back reflections from optical components in the beam path into the laser diode are attentuated with an optical Faraday isolator (FI). The laser beam is coupled into a polarization maintaining single mode optical fiber1 for spatial mode cleaning. Both fiber end facets are angle cleaved to avoid sur- faces which are perpendicular to the beam path, and which might lead to unstable power of the transmitted laser beam due to the build up of unrequested Fabry-Perot cavities.
Figure 4.4: Setup of the Michelson interferometer
After outcoupling from the fiber, the beam is collimated and the polarization is cleaned with a λ/2 plate (WP) and a polarizing beam splitter (PBS) such that the polarization of the transmitted beam is vertically oriented. The power is split at the 50/50 beam splitter in the center of the interferometer, and one light beam is focussed onto the membrane with a lens of f = 100 mm. The beam waist on the membrane is approximately w0 = 200 µm. The membrane is mounted on a solid
aluminum cuboid in a vacuum chamber which consists of a six way cross (CF40) which is permanently pumped with an ion pump to a pressure p < 1×106 mbar. A hole at the membrane position in the cuboid and two broadband antireflection coated windows mounted at opposite flanges allow optical access to the membrane from both sides. The membrane frame is UV glued to the cuboid at one corner only in order to avoid bending of the frame due to shrinking of the glue volume during the curing procedure, which might impose unwanted stress onto the membrane. Reflec- tions from the windows into the beam path are avoided by rotating the membrane by 10◦ with respect to the windows. A neutral optical density filter in the reference beam path of the interferometer is introduced to match the powers of the interfering beams. The mirror in the reference arm is a gold coated glass plate which is glued onto a low voltage multi stack piezo, which allows to vary the length of this arm by one wavelength per 10V.
The interfered amplitudes of the overlapped beams are measured with an amplified photo detector (PD) as shown in Fig. 4.5. We use a BPW34 with a reverse bias voltage of 15 V and perform a current to voltage conversion with the operational amplifier OP 37, which combines low noise and a gain-bandwidth product which is sufficient for our purposes. The photodiode circuit provides three different outputs:
4.2 Extension of the atom chip setup 63
Figure 4.5: Amplification and filtering circuitry of the high sensitivity photo detector.
(1) right after the current to voltage conversion (general out), (2) a low pass filtered signal with subsequent operational amplifier OP 27 (DC out), (3) a high pass fil- tered signal with subsequent operational amplifier OP 27, providing a gain of 5 to compensate for the drop of the signal level due to filtering (AC out).
To reduce pickup noise it is essential to build the circuit and the photodetector into a massive aluminum box. The wires which guide the current from the photodiode to the current to voltage conversion amplifier should be as short as possible2. The adjustment of the resistor in the feedback arm of the operational amplifier follows a trade off between sensitivity and bandwidth. In order to measure the bandwidth of the photodiode, a laser beam which is incident on the photodiode is switched on, and the rise time of the signal at the photodiode is measured at the respective output. The bandwidth is given by the inverse of the time that has elapsed when the signal has reached 63 % of the steady state level. We use the solid state switch in the AOM controller3 with a switching time of a few tens of ns, which is small in comparison to the timescale on which the signal rises. The bandwidth of the photodiode is measured and adjusted to 1 MHz at the general output .The rms noise at the general output is 3.5×10−5 of the full range (12 V ).
Operation of the Michelson interferometer
The light beams which are incident on the photodiode are overlapped carefully in order to achieve a maximum modulation of the interfering beams for a variation of
2A trial to mount the photodiode in a separate box and to guide the current with LEMO con-
nectors and shielded cables to the current to voltage conversion amplifier resulted in significant pick up noise.
3AOM
Figure 4.6: Photograph of the Michelson interferometer.
the relative optical path length (L2−L1), and to achieve a minimum power modu- lation for destructive interference as shown in Fig. 4.3. In the alignment procedure the power of the beam which is reflected from the membrane and coupled back into the optical fiber is monitored with the photodiode (PD1). Maximum power indicates that the beam axis is perpendicular to the membrane, i.e. the incoming and outbound beams are overlapping. A second condition is that the beam hits the center of the membrane. These two requirements are fulfilled simultaneously by aligning the two steering mirrors M3 and M4. In order to overlap the two light beams incident on the photodiode (PD), the power modulation is maximized while tapping the vacuum chamber with the membrane which induces variations of the relative optical path length (L2−L1) of more thanλ/2. The goal of the alignment
procedure is that the interference pattern is only one spot, and that the brightness changes common mode over the whole area when (L2−L1) is varied.
Mechanical vibrations which induce arbitrary variations of the relative optical path length (L2 − L1) are minimized by choosing stable mechanical components and
mounting the interferometer as compact as possible. All relevant optical compo- nents are mounted on 1/2 inch posts and in stable holders4. The soft spot of the
interferometer is the mechanical mounting of the membrane in the vacuum chamber which is susceptible to acoustic vibrations.
4.2 Extension of the atom chip setup 65
The relative path length (L2−L1) is actively stabilized with a PI regulator5 which
acts on the piezo mirror in the reference arm of the interferometer. To take out acoustic vibrations, it is sufficient to stabilize the interferometer to frequencies in the kHz range. In particular, the stabilization should not work at the frequencies close to the membrane eigenfrequencies. To ensure this, we use the DC out of the photodiode PD and adjust the cutoff of the low pass filter (LP) to 13 kHz. The maximum signal level of the DC output is adjusted with the operational amplifier OPA2 in order to scale to the signal level that the PI regulator requires. The signal from the DC output is galvanically isolated and input as actual value to the PI regulator. The proportional and integral part of the output are summed with an operational amplifier OP 27 in a one to one summing amplifier configuration, and fed to the piezo (PIstab). The set value is adjusted with a potentiometer such that the level of the DC output is stabilized to the steepest slope.
Figure 4.7: The thermal motion of the membrane allows to scale the ordinate.
The interferometer is calibrated with the thermal motion of the fundamental mode of the membrane oscillator which is measured with an oscilloscope6at the AC output
of the photodiode PD. An FFT is applied to the time trace, and a Lorentzian is fitted to the peak of the power fluctuations of the photodiode current. The integrated area
5LB
5, Toni Scheich 6LeCroy waveRunner 44Xi
below the Lorentzian is the squared amplitude of the membrane in units [m2]. The
amplitude is taken as the amplitude of a harmonic oscillator and set equal tokBT /2
in order to calculate a scaling factor for the ordinate. The effective mass of the SiN membrane7 with dimensions 0.5 mm×0.5 mm×50nm is m
ef f = 1.1×10−8 g. With
the calculated rms amplitude of 1.6× 10−11m, the ordinate is rescaled as shown in Fig. 4.7 and the noise floor is found to be at 2.9×10−14 m/√Hz. A relative
length change of (L2 −L1) = 2.9×10−14m results in a change of the current I of δI = 2.97×107 e/s, if the interferometer is locked to the steepest slope. This coin- cides with the shotnoise fluctuations of the photodiode current of 70µA measured in a bandwidth of 1 Hz, which is estimated with equation 4.53 to beσI = 2.96×107 e/s.
This analysis shows, that the performance of the interferometer is limited by shot- noise in the photodetector.